Rational trigonometry

From Wikipedia, the free encyclopedia

Divine Proportions: Rational Trigonometry to Universal Geometry is a book by Norman J. Wildberger, an outspoken critic of traditional mathematics, presenting his reformulation of trigonometry. Wildberger holds a Ph.D. in mathematics from Yale University, and taught at Stanford University from 1984 to 1986 and at the University of Toronto from 1986 to 1989; he is currently an associate professor of mathematics at the University of New South Wales, Australia.

[edit] Quadrance and spread

Instead of distance and angle, rational trigonometry uses as its fundamental units quadrance (square of distance) and spread (square of sine of angle).^[1] This choice of variables enables calculations to produce output results whose complexity matches that of the input data. For instance, in a typical trigonometry problem if rational numbers are assigned to all quadrances and spreads, then the calculated results will be rational numbers (or roots of rational numbers).

This rationality is obtained at the expense of linearity. Unlike the traditional distance and angle units, doubling or halving a quadrance or spread does not double or halve a length or a rotation. Similarly, the sum of two lengths or rotations is not the sum of their individual quadrances or spreads.

For distinction, Wildberger refers to the traditional trigonometry as classical trigonometry. It is otherwise broadly based on Cartesian analytic geometry, with a point defined as an ordered pair (x, y) and a line as a general linear equation

$Ax + By + C = 0.\,$

The mathematics of rational trigonometry is, applications aside, a special instance of the description of geometry in terms of linear algebra (using rational methods such as dot products and quadratic forms), but students who are first learning trigonometry are often not taught about the use of linear algebra in geometry. Changing this state of affairs is a stated aim of Wildberger's book (to paraphrase his comments).

[edit] Trigonometry over finite fields

Rational trigonometry makes it possible to do trigonometry over finite fields in the same way as over the field of real numbers.

[edit] Quadrance

The quadrance is the square of the distance.^[1]

Quadrance and distance are concerned with the separation of points. Quadrance differs from standard distance in that it squares the distance. Most immediately, this means that calculating the distance (or, more accurately, quadrance) between two points in 2-dimensional space is easier, as there is no need to find the square root of the sum of the squares of the differences in the x and y coordinates.

In the (x, y)-plane, the quadrance Q(A₁, A₂) for the points A₁ and A₁ is defined as

$Q(A_1, A_2) = (x_2 - x_1)^2 + (y_2 - y_1)^2.\,$ ^[1]

[edit] Spread

Suppose ℓ₁ and ℓ₂ intersect at the point A. Let C be the foot of the perpendicular from B to ℓ₂. Then the spread is s = Q/R.

The spread of two lines can be measured in four equivalent positions.

Spread, a measure of the separation of lines, is a dimensionless number in the range [0, 1]. The value of the spread is the square of the sine of the angle.^[1]

Suppose two lines, ℓ₁ and ℓ₂, intersect at the point A as shown at right. Choose a point B ≠ A on ℓ₁ and let C be the foot of the perpendicular from B to ℓ₂. Then the spread s is

$s(\ell_1, \ell_2) = \frac{Q(B, C)}{Q(A, B)} = \frac{Q}{R}.$ ^[1]

[edit] Spread compared to angle

In rational trigonometry, spread is a fundamental concept, somewhat but not precisely corresponding to the concept in traditional geometry of angle. Spread describes a relationship between two lines, whereas angle describes a relationship between two rays emanating from a common point.^[1]

The spread is the square of the sine of the angle.

Degree	Radian	Spread
0	0	0
30	(1/6)π	1/4
45	(1/4)π	1/2
60	(1/3)π	3/4
90	(1/2)π	1
120	(2/3)π	3/4
135	(3/4)π	1/2
150	(5/6)π	1/4
180	π	0

Spread is not proportional to degrees or radians, and has a period of 180 degrees (π radians).

[edit] Laws of rational trigonometry

Wildberger states that there are five basic laws in rational trigonometry and these laws can be easily verified using high-school level mathematics. Some are equivalent to standard trigonometrical formulae with the variables expressed as quadrance and spread.^[1]

[edit] Triple quad formula

The three points $A 1, A 2, A 3$ are collinear precisely when:

$(Q_{1} + Q_{2} + Q_{3})^2 = 2(Q_{1}^{2} + Q_{2}^{2} + Q_{3}^{2}).\,$

This is equivalent to using Heron's formula, the condition for collinearity being that the triangle formed by the three points has zero area.

It can either be proved by analytic geometry (the preferred means within rational trigonometry) or derived from Heron's formula, using the condition for collinearity that the triangle formed by the three points has zero area.

Proof

Illustration of nomenclature used in the proof.

The line $AB\,$ has the general form:

$ax + by + c = 0\,$

where the (non-unique) parameters $a\,$ , $b\,$ and $c\,$ , can be expressed in terms of the coordinates of points $A\,$ and $B\,$ as:

$a = A_y - B_y\,$

$b = B_x - A_x\,$

$c = A_xB_y - A_yB_x\,$

so that, everywhere on the line:

$(A_y - B_y)x + (B_x - A_x)y + (A_xB_y - A_yB_x) = 0.\,$

But the line can also be specified by two simultaneous equations in a parameter $t\,$ , where $t = 0\,$ at point $A\,$ and $t = 1\,$ at point $B\,$ :

$x = (B_x - A_x)t + A_x\,$ and $y = (B_y - A_y)t + A_y\,$

or, in terms of the original parameters:

$x = bt + A_x\,$ and $y = -at + A_y.\,$

If the point $C\,$ is collinear with points $A\,$ and $B\,$ , there exists some value of $t\,$ (for distinct points, not equal to 0 or 1), call it $\lambda\,$ , for which these two equations are simultaneously satisfied at the coordinates of the point $C\,$ , such that:

$C_x = b\lambda\ + A_x$ and $C_y = -a\lambda\ + A_y.\,$

Now, the quadrances of the three line segments are given by the squared differences of their coordinates, which can be expressed in terms of $\lambda\,$ :

$\begin{matrix}Q(AB) & \equiv & (B_x - A_x)^2 + (B_y - A_y)^2 \\ \ & = & b^2 + (-a)^2 \\ \ & = & a^2 + b^2\end{matrix}$

$\begin{matrix}Q(BC) & \equiv & (C_x - B_x)^2 + (C_y - B_y)^2 \\ \ & = & ((b\lambda\ + A_x) -B_x)^2 + ((-a\lambda\ + A_y) - B_y)^2 \\ \ & = & (b\lambda\ + (A_x -B_x))^2 + (-a\lambda\ + (A_y - B_y))^2\\ \ & = & (b\lambda\ + (-b))^2 + (-a\lambda\ + a)^2\\ \ & = & b^2(\lambda\ - 1)^2 + a^2(-\lambda\ + 1)^2 \\ \ & = & b^2(\lambda\ - 1)^2 + a^2(\lambda\ - 1)^2\\ \ & = & (a^2 + b^2)(\lambda\ - 1)^2\end{matrix}$

$\begin{matrix}Q(AC) & \equiv & (C_x - A_x)^2 + (C_y - A_y)^2 \\ \ & = & ((b\lambda\ + A_x) - A_x)^2 + ((-a\lambda\ + A_y) - A_y)^2 \\ \ & = & (b\lambda\ + A_x - A_x)^2 + (-a\lambda\ + A_y - A_y)^2\\ \ & = & (b\lambda)^2 + (-a\lambda)^2\\ \ & = & b^2\lambda^2 + (-a)^2\lambda^2\\ \ & = & b^2\lambda^2 + a^2\lambda^2\\ \ & = & (a^2 + b^2)\lambda^2\end{matrix}$

where use was made of the fact that $(-\lambda\ + 1)^2 = (\lambda\ - 1)^2$ .

Substituting these quadrances into the equation to be proved:

$(Q(AB) + Q(BC) + Q(AC))^2 = 2(Q(AB)^{2} + Q(BC)^{2} + Q(AC)^{2})\,$

$((a^2 + b^2) + (a^2 + b^2)(\lambda\ - 1)^2 + (a^2 + b^2)\lambda^2)^2 = 2((a^2 + b^2)^2 + ((a^2 + b^2)(\lambda\ - 1)^2)^2 + ((a^2 + b^2)\lambda^2)^2)\,$

$(a^2 + b^2)^2(1 + (\lambda\ - 1)^2 + \lambda^2)^2 = 2(a^2 + b^2)^2(1 + ((\lambda\ - 1)^2)^2 + (\lambda^2)^2)\,$

Now, if $A\,$ and $B\,$ represent distinct points, such that $(a^2 + b^2)\,$ is not zero, we may divide both sides by $Q(AB)^2 = (a^2 + b^2)^2\,$ :

$(1 + \lambda^2 -2\lambda\ + 1 + \lambda^2)^2 = 2(1 + (\lambda^2 -2\lambda\ + 1)^2 + \lambda^4)\,$

$(2\lambda^2 - 2\lambda\ + 2)^2 = 2(1 + \lambda^4 - 2\lambda^3 + \lambda^2 - 2\lambda^3 + 4\lambda^2 - 2\lambda + \lambda^2 - 2\lambda + 1 + \lambda^4)\,$

$4(\lambda^2 - \lambda\ + 1)^2 = 2(2\lambda^4 - 4\lambda^3 + 6\lambda^2 - 4\lambda + 2)\,$

$4(\lambda^4 - \lambda^3 + \lambda^2 - \lambda^3 + \lambda^2 - \lambda + \lambda^2 - \lambda + 1) = 4(\lambda^4 - 2\lambda^3 + 3\lambda^2 - 2\lambda + 1)\,$

$\lambda^4 - 2\lambda^3 + 3\lambda^2 - 2\lambda + 1 = \lambda^4 - 2\lambda^3 + 3\lambda^2 - 2\lambda + 1\,$

Q.E.D.

[edit] Pythagoras' theorem

The lines $A 1 A 3$ and $A 2 A 3$ are perpendicular precisely when:

$Q_{1} + Q_{2}= Q_{3}.\,$

This is equivalent to the Pythagorean theorem.

There are many classical proofs of Pythagoras' theorem; this one is framed in the terms of rational trigonometry.

The spread of an angle is the square of its sine. Given the triangle ABC with a spread of 1 between sides AB and AC,

$Q(AB) + Q(AC) = Q(BC)\,$

where Q is the "quadrance", i.e. the square of the distance.

Proof

Illustration of nomenclature used in the proof.

Construct a line AD dividing the spread of 1, with the point D on line BC, and making a spread of 1 with DB and DC. The triangles ABC, DBA and DAC are similar (have the same spreads but not the same quadrances).

This leads to two equations in ratios, based on the spreads of the sides of the triangle:

$s_C = \frac{Q(AB)}{Q(BC)} = \frac{Q(BD)}{Q(AB)} = \frac{Q(AD)}{Q(AC)}.$

$s_B = \frac{Q(AC)}{Q(BC)} = \frac{Q(DC)}{Q(AC)} = \frac{Q(AD)}{Q(AB)}.$

Now in general, the two spreads resulting from dividing a spread into two parts, as line AD does for spread CAB, do not add up to the original spread since spread is a non-linear function. So we first prove that dividing a spread of 1, results in two spreads that do add up to the original spread of 1.

For convenience, but with no loss of generality, we orient the lines intersecting with a spread of 1 to the coordinate axes, and label the dividing line with coordinates $(x 1, y 1)$ and $(x 2, y 2)$ . Then the two spreads are given by:

$s_1 = \frac{(x_2 - x_2)^2 + (y_2 - y_1)^2}{(x_2 - x_1)^2 + (y_2 - y_1)^2} = \frac{(y_2 - y_1)^2}{(x_2 - x_1)^2 + (y_2 - y_1)^2},$

$s_2 = \frac{(x_2 - x_1)^2 + (y_2 - y_2)^2}{(x_2 - x_1)^2 + (y_2 - y_1)^2} = \frac{(x_2 - x_1)^2}{(x_2 - x_1)^2 + (y_2 - y_1)^2}.$

Hence:

$s_1 + s_ 2 = \frac{(x_2 - x_1)^2 + (y_2 - y_1)^2}{(x_2 - x_1)^2 + (y_2 - y_1)^2} = 1.\,$

So that:

$s_C + s_B = 1.\,$

Using the first two ratios from the first set of equations, this can be rewritten:

$\frac{Q(AB)}{Q(BC)} + \frac{Q(AC)}{Q(BC)} = 1.\,$

Multiplying both sides by $Q (B C)$ :

$Q(AB) + Q(AC) = Q(BC).\,$

Q.E.D.

[edit] Spread law

For any triangle $\overline{A_{1} A_{2} A_{3}}$ with non zero quadrances:

$\frac{s_{1}}{Q_{1}}=\frac{s_{2}}{Q_{2}}=\frac{s_{3}}{Q_{3}}.\,$

This is equivalent to the law of sines.

[edit] Cross law

For any triangle $\overline{A_{1} A_{2} A_{3}}$ define the cross c₃ as c₃ = 1 − s₃. Then:

$(Q_1 + Q_2 - Q_3)^2 = 4Q_1 Q_2 c_3.\,$

This is equivalent to the law of cosines.

[edit] Triple spread formula

For any triangle $\overline{A_1 A_2 A_3},$

$(s_1 + s_2 + s_3)^2 = 2(s_1^2 + s_2^2 + s_3^2) + 4s_1 s_ 2 s_ 3 .\,$

This corresponds (roughly) to the angle sum formulae for sine and cosine.

[edit] Calculating quadrance and spread

Spread calculation

Given the coordinates of two points (x₁, y₁) and (x₂, y₂), the quadrance between them is

$Q = (x_2 -x_1)^2 + (y_2 - y_1)^2.\,$

Given two line segments A₁A₃ and A₁A₃ (forming an angle at point A₃, the spread between them is

$\begin{align} s & = 1 - \frac{(Q_1 + Q_2 - Q_3 )^2}{4Q_1 Q_2} \\[6pt] & = \frac{ 2Q_1 Q_2 + 2Q_1 Q_3 + 2Q_2 Q_3 - Q_1^2 - Q_2^2 - Q_3^2}{4Q_1 Q_2} \end{align}$

where Q₁ is the quadrance of the side opposite the vertex A₁, Q₂ is the quadrance of the side opposite the vertex A₂, Q₃ is the quadrance of the side opposite the vertex A₃.

Given the coordinates of two points on each of two lines $(x 11, y 11)$ , $(x 12, y 12)$ and $(x 21, y 21)$ $(x 22, y 22)$ , the spread s between them can be calculated as:

$s = \frac{((x_{12} - x_{11})(y_{22} - y_{21}) - (x_{22} - x_{21})(y_{12} - y_{11}))^2}{((x_{12}-x_{11})^2+(y_{12}-y_{11})^2) ((x_{22}-x_{21})^2+(y_{22}-y_{21})^2)}$

or

$s = \frac{(\Delta x_1 \, \Delta y_2 - \Delta x_2\, \Delta y_1)^{2}}{Q_1 Q_2}.\,$

If the lines described by the points emanate from or are shifted to the origin by subtracting the coordinates of the first point from each line, as illustrated on the right, the computation simplifies to

$s = \frac{(x_1 y_2 - x_2 y_1)^2}{Q_1 Q_2}.\,$

[edit] See also

Spread polynomials

[edit] References

^ ^a ^b ^c ^d ^e ^f ^g Wildberger, Norman J. (2007), "A Rational Approach to Trigonometry", Math Horizons (Washington, DC: Mathematical Association of America) November 2007: 16–20, ISSN 1072-4117

Wildberger's rational trigonometry site, including downloadable papers and sections of his book
A comparison of classical and rational trigonometry
Alexander Bogomolny (2007). "A brief introduction to Rational Trigonometry". Cut-the-knot. http://www.cut-the-knot.org/pythagoras/RationalTrig/CutTheKnot.shtml.

[edit] External links

[horizons-0] ^ ^a ^b ^c ^d ^e ^f ^g Wildberger, Norman J. (2007), "A Rational Approach to Trigonometry", Math Horizons (Washington, DC: Mathematical Association of America) November 2007: 16–20, ISSN 1072-4117

[1]

Rational trigonometry

From Wikipedia, the free encyclopedia

Contents

[edit] Quadrance and spread

[edit] Trigonometry over finite fields

[edit] Quadrance

[edit] Spread

[edit] Spread compared to angle

[edit] Laws of rational trigonometry

[edit] Triple quad formula

[edit] Pythagoras' theorem

[edit] Spread law

[edit] Cross law

[edit] Triple spread formula

[edit] Calculating quadrance and spread

[edit] See also

[edit] References

[edit] External links

Views

Personal tools

Navigation

Search

Interaction

Toolbox

Languages